Linear Algebra by Shilov: A Comprehensive Guide for Undergraduates | Amazon Link

[micromass]

• Author: G.E. Shilov
• Title: Linear Algebra
• Amazon link https://www.amazon.com/dp/048663518X/?tag=pfamazon01-20
• Prerequisities: Being acquainted with proofs and rigorous mathematics. Knowing what matrices and determinants are, is also helpful.
• Level: Undergrad

Table of Contents:

[LIST]
[*] Determinants
[LIST]
[*] Number Fields
[*] Problems of the Theory or Systems or Linear Equations
[*] Determinants of Order [itex]n[/itex]
[*] Properties of Determinants
[*] Cofactors and Minors
[*] Practical Evaluation of Determinants
[*] Cramer's Rule
[*] Minors of Arbitrary Order. Laplace's Theorem
[*] Linear Dependence between Columns
[*] Problems
[/LIST][*] Linear Spaces
[LIST]
[*] Definitions
[*] Linear Dependence
[*] Bases, Components, Dimension
[*] Subspaces 
[*] Linear Manifolds 
[*] Hyperplanes 
[*] Morphisms of Linear Spaces
[*] Problems
[/LIST][*] Systems of Linear Equations
[LIST]
[*] More on the Rank of a Matrix 
[*] Nontrivial Compatibility of a Homogeneous Linear System 
[*] The Compatibility Condition for a GeneraI Linear System 
[*] The Generai Solution of a Linear System 
[*] Geornetric Properties of the Solution Space 
[*] Methods for CaJculating the Rank or a Matrix 
[*] Problems
[/LIST][*] Linear Functions of a Vector Argument
[LIST]
[*] Linear Forrns
[*] Linear Operators 
[*] Sums and Products of Linear Operators 
[*] Corresponding Operations on Matrices 
[*] Further Properties of Matrix Multiplication 
[*] The Range and Null Space or a Linear Operator 
[*] Linear Operators Mapping a Space [itex]\mathbb{K}_n[/itex] into Itself 
[*] Invariant Subspaces 
[*] Eigenvectors and Eigenvalues 
[*] Problems
[/LIST][*] Coordinate Transformations
[LIST]
[*] Transformation to a New Basis
[*] Consecutive Transformations 
[*] Transformation of the Components of a Vector 
[*] Transformation of the Coefficients of a Linear Form 
[*] Transformation of the Matrix of a Linear Operator 
[*] Tensors 
[*] Problems 
[/LIST][*] The Canonical Form of the Matrix of a Linear Operator
[LIST]
[*] Canonical Form of the Matrix of a Nilpotent Operator
[*] Algebras. The Algebra of Polynomials 
[*] Canonical Form of the Matrix of an Arbitrary Operator 
[*] Elementary Divisors 
[*] Further Implications 
[*] The Real Jordan Canonical F orrn 
[*] Spectra, Jets and Polynomials 
[*] Operator Functions and Their Matrices 
[*] Problems
[/LIST][*] Bilinear and Quadratic Forms
[LIST]
[*] Bilinear Forms
[*] Quadratic Forms 
[*] Reduction of a Quadratic Form to Canonical Form 
[*] The Canonical Basis of a Bilinear Form 
[*] Construction of a Canonical Basis by Jacobi's Method 
[*] Adjoint Linear Operators 
[*] Isomorphism of Spaces Equipped with a Bilinear Form 
[*] Multilinear Forms
[*] Bilinear and Quadratic Forms in a Real Space
[*] Problems
[/LIST][*] Euclidean Spaces
[LIST] 
[*] Introduction
[*] Definition of a Euclidean Space
[*] Basic Metric Concepts 
[*] Orthogonal Bases 
[*] Perpendiculars 
[*] The Orthogonalization Theorem 
[*] The Gram Determinant 
[*] Incompatible Systems and the Method of Least Squares 
[*] Adjoint Operators and Isometry 
[*] Problems
[/LIST][*] Unitary Spaces
[LIST]
[*] Hermitian Forms
[*] The Scalar Product in a Complex Space 
[*] Normal Operators 
[*] Applications to Operator Theory in Euclidean Space 
[*] Problems
[/LIST][*] Quadratic Forms in Euclidean and Unitary Spaces
[LIST] 
[*]Basic Theorem on Quadratic Forms in a Euclidean Space
[*] Extremal Properties of a Quadratic Form 
[*] Simultaneous Reduction of Two Quadratic Forms 
[*] Reduction of the Generai Equation of a Quadric Surface 
[*] Geometric Properties of a Quadric Surface
[*] Analysis of a Quadric Surface from Its GeneraI Equation 
[*] Hermitian Quadratic Forms 
[*] Problems 
[/LIST][*] Finite-dimensional Algebras and Their Representations
[LIST]
[*] More on Algebras
[*] Representations of Abstract Algebras 
[*] Irreducible Representations and Schurs Lemma 
[*] Basic Types or Finite-Dimensional Algebras 
[*] The Left Regular Representation of a Simple Algebra 
[*] Structure of Simple Algebras 
[*] Structure of Semisimple Algebras 
[*] Representations or Simple and Semisimple Algebras 
[*] Some Further Results 
[*] Problems
[/LIST][*] Appendix: Categories of Finite-Dimensional Spaces
[LIST]
[*] Introduction 
[*] The Case or Complete Algebras 
[*] The Case of One-Dimensional Algebras 
[*] The Case of Simple Algebras 
[*] The Case of Complete Algebras of Diagonal Matrices 
[*] Categories and Direct Sums
[/LIST][*] Hints and Answers[*] Bibliography[*] Index
[/LIST]

 

[A. Bahat]

I'm not a huge fan of this book. While it has some nice explanations, the coordinates are overwhelming! Sums and bases and indices, oh my! E.g. determinants make up the first chapter, which strikes me as odd, and as such, the chapter ends up very computational--he hasn't even defined a linear map at this point.

For an introduction to linear algebra, I'd prefer the relevant chapters of Artin, or maybe Hoffman/Kunze.

 

[PKDfan]

I'm learning linear algebra from this book (meaning no previous exposure to the subject) and so far I love it! I like how he has computational material/problems in additional to the very high-level abstract stuff. Makes the book feel very "balanced". Granted, I haven't gotten that far into it, but so far it's been very easy to learn from.